| L(s) = 1 | − 0.554i·3-s + 4.57·5-s − 2.64i·7-s + 8.69·9-s + 15.7i·11-s − 8.57·13-s − 2.53i·15-s + 28.3·17-s + 6.33i·19-s − 1.46·21-s − 31.0i·23-s − 4.05·25-s − 9.81i·27-s + 0.846·29-s + 21.6i·31-s + ⋯ |
| L(s) = 1 | − 0.184i·3-s + 0.915·5-s − 0.377i·7-s + 0.965·9-s + 1.43i·11-s − 0.659·13-s − 0.169i·15-s + 1.66·17-s + 0.333i·19-s − 0.0698·21-s − 1.35i·23-s − 0.162·25-s − 0.363i·27-s + 0.0291·29-s + 0.697i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.210890257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210890257\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 + 0.554iT - 9T^{2} \) |
| 5 | \( 1 - 4.57T + 25T^{2} \) |
| 11 | \( 1 - 15.7iT - 121T^{2} \) |
| 13 | \( 1 + 8.57T + 169T^{2} \) |
| 17 | \( 1 - 28.3T + 289T^{2} \) |
| 19 | \( 1 - 6.33iT - 361T^{2} \) |
| 23 | \( 1 + 31.0iT - 529T^{2} \) |
| 29 | \( 1 - 0.846T + 841T^{2} \) |
| 31 | \( 1 - 21.6iT - 961T^{2} \) |
| 37 | \( 1 - 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 66.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 44.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 5.80iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 52.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 89.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52500521768296817441030782393, −9.944101634430749643899017686756, −9.476755872036026571829104664380, −7.906997579183338364909492369344, −7.23078080179236468984068084428, −6.27951379880384951819178106542, −5.08735503275976674006816356629, −4.14580575428081526446981353702, −2.45993811821797887392246656966, −1.30575056083804086538103561115,
1.18956589516220750860282897463, 2.68913707392263347276430155669, 3.93661784930967351101725245724, 5.44872936211707588693954487190, 5.87362403490258308848048463811, 7.23915907569959408646546069899, 8.133159928632185570686677760332, 9.475344897256832940859954946960, 9.722697192078246971607071333515, 10.80514796589840256590226587892
